Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified learning theory for these settings has been lacking. To start addressing this gap, we introduce a novel convolutional learning framework for possibly infinite-dimensional signals supported on a manifold. Namely, we use the connection Laplacian associated with a Hilbert bundle as a convolutional operator, and we derive filters and neural networks, dubbed as \textit{HilbNets}. We make HilbNets and, more generally, the convolution operation, implementable via a two-stage sampling procedure. First, we show that sampling the manifold induces a Hilbert Cellular Sheaf, a generalized graph structure with Hilbert feature spaces and edge-wise coupling rules, and we prove that its sheaf Laplacian converges in probability to the underlying connection Laplacian as the sampling density increases. Notably, this result is a generalization to the infinite-dimensional bundle setting of the Belkin \& Niyogi \cite{BELKIN20081289} convergence result for the graph Laplacian to the manifold Laplacian, a theoretical cornerstone of geometric learning methods. Second, we discretize the signals and prove that the discretized (implementable) HilbNets converge to the underlying continuous architectures and are transferable across different samplings of the same bundle, providing consistency for learning. Finally, we validate our framework on synthetic and real-world tasks. Overall, our results broaden the scope of geometric learning as a whole by lifting classical Laplacian-based frameworks to settings where the signal at each point lives in its own Hilbert space.
Scaling Graph Reasoning in Language Models with Learnable Graph Positional Encodings
Javier Porras-Valenzuela, Arush Arora, Alejandro Ribeiro
AI 速览：作者提出用可学习的、置换等变的图位置编码调制语言模型注意力，并以GNN加Graph Transformer计算编码，从而避免序列化邻接表、把上下文负担降到O(N)。
Solving reasoning tasks supported on graphs typically requires providing a Language Reasoning Model (LRM) with a serialized description of the graph. This approach results in the input length growing with the number of edges, which limits the size of the problems that can be solved. Furthermore, even when graphs fit in context, performance sharply decays as the graphs grow. We propose to address this by augmenting LRMs with scalable and permutation equivariant graph positional encodings that modulate attention logits. An additional post-fusion-style module incorporates structural information into hidden states. We compute these positional encodings using a GNN followed by a Graph Transformer. This eliminates the need for serialized adjacencies, reducing the context burden to $O(N)$. We jointly fine-tune the positional encoder and the LRM LoRA adapters using optimal solutions generated by graph-theoretic solvers, resulting in a computationally efficient method. Trained on small graphs, the model exhibits improved size-generalization across graph-theoretic tasks on graphs with up to 1,000 nodes, and produces fewer tokens than a conventional LRM. Additionally, we observe that our model remains robust to node relabelings.
Antonio Pariente, Ignacio Boero, Nikolai Matni, Alejandro Ribeiro
World models jointly learn latent representations and dynamics that predict how high-dimensional observations evolve under actions. In this work, we propose a JEPA-style world model in which, rather than learning arbitrary latent dynamics, we restrict them to follow a bilinear parameterization. This structure enables efficient planning and control while shifting the modeling burden onto the encoder, encouraging richer representations that expose the controllable geometry of the system. In particular, this structured parameterization allows us to structurally enforce action recoverability, thereby preventing representation collapse by construction. Although prescribing a bilinear parametrization may appear restrictive, we show that a broad class of nonlinear dynamical systems admits a transformation under which the dynamics become bilinear. Empirically, we show across standard 2D and 3D control tasks that representations with bilinear-parameterized dynamics can be learned directly from high-dimensional observations, reducing planning time by nearly three orders of magnitude while retaining or even improving control accuracy. We also propose more demanding regimes of longer-horizon planning and real-time control, and demonstrate that our method succeeds in both, moving JEPA-style world models beyond short-horizon offline planning.